So I have soul trap on my bow that says "If target dies within 4 seconds, fills up a soul gem". Killing within 4 seconds with a bow is not a big deal when I can shot a lot of my targets, but I'm getting a message that my crystal is not large enough. I have a bunch of empty crystals in my inventory, bought biggest one I could find (great i think) and even have a black one. Do I need to "install"empty crystals to start capturing souls?

Petty > Lesser > Common > Greater > Grand. Whatever you're trying to catch is probably has a grand soul. Or all the soul gems in your inventory are full.

I find this strange as well. I tried capturing the soul of a common guard in a dungeon with an empty grand soul gem in my inventory, only for a message to say that I don't have a big enough gem. Really? How much bigger need this gem be?

You cannot soultrap humanoid NPC' s unless its with a black soul gem.

For human-esque souls you need black soul gems

Buy at least one of every size soul gem and try to capture the soul. The soul will only go into a soul gem that is the same size as itself, so a common soul will only go into a common gem, and nothing but a grand soul can go into a grand gem. You also can't have more than one soul in one gem, so you can't fill a grand soul gem up gradually with lesser souls. If you plan on capturing souls frequently, you should stock up on empty gems. If your character is a thief, or if you just explore regularly, you'll rarely have to actually buy more empty gems as you'll just find them out in the world.

Or, get azura's star from a certain quest.

So for non-black gems do I hunt animals if I can't use humanoids?

anything non human(or beast race) really, so trolls, wisp mothers, and whatnot are all fair game

Basically anything that isn't Humanoid-NPC, yes. Everything from Deer, to MudCrab to spideys and the like.